Numerical Approximation of Riesz-Feller Operators on \mathbb R

2024/01/13 by Carlota M. Cuesta, Cuesta, Carlota M., de la Hoz, Francisco +2
Mathematics · #26A33 #33C05 #65T50 #Differential Equations and Boundary Problems #FOS: Mathematics #Mathematical functions and polynomials #Numerical Analysis (math.NA) #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.2401.07140

openalex publication_date 2024/01/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we develop an accurate pseudospectral method to approximate numerically the Riesz-Feller operator D_γ^α on \mathbb R, where α∈(0,2), and |γ|≤min\α, 2 - α\. This operator can be written as a linear combination of the Weyl-Marchaud derivatives D^α and D^α, when α∈(0,1), and of ∂_xD^α-1 and ∂_xD^α-1, when α∈(1,2). Given the so-called Higgins functions λ_k(x) = ((ix-1)/(ix+1))^k, where k∈\mathbb Z, we compute explicitly, using complex variable techniques, D^α[λ_k](x), D^α[λ_k](x), ∂_xD^α-1[λ_k](x), ∂_xD^α-1[λ_k](x) and D_γ^α[λ_k](x), in terms of the Gaussian hypergeometric function ₂F₁, and relate these results to previous ones for the fractional Laplacian. This enables us to approximate D^α[u](x), D^α[u](x), ∂_xD^α-1[u](x), ∂_xD^α-1[u](x) and D_γ^α[u](x), for bounded continuous functions u(x). Finally, we simulate a nonlinear Riesz-Feller fractional diffusion equation, characterized by having front propagating solutions whose speed grows exponentially in time.

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